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5
votes
3answers
519 views

Matrices generating non-discrete subgroups of SL(2,R)

Jorgensen's inequality $\mid \left(Tr\left(A\right)\right)^2-4\mid+\mid Tr\left[A,B\right]-2\mid\ge 1$ gives a necessary condition for two matrices A,B to generate a discrete subgroup of SL(2,R). Are ...
3
votes
4answers
599 views

Softwares for drawing hyperbolic surfaces , closed, with boundaries or with punctures ?

In a paper I am in the process of writing in LaTeX, I need to draw and incorporate some diagrams of hyperbolic surfaces in my LaTeX document. Is there any software I can use to draw hyperbolic ...
4
votes
1answer
334 views

Does the fundamental group of a surface have rigid subgroups?

Suppose $\Gamma$ is the fundamental group of a closed, oriented surface $S$. Let $B$ be a finitely generated, infinite index subgroup of $\Gamma$, and let $\Gamma_B$ be the compact core of the ...
0
votes
1answer
166 views

Uniqueness of distance realizing geodesic in hyperbolic surface. [duplicate]

Possible Duplicate: Hyperbolic surfaces Given a hyperbolic surface S with geodesic boundary. Let a and b be two distinct simple closed geodesic boundaries. Does there exist a unique distance ...
3
votes
1answer
277 views

Pleated surfaces do not curl up too much

Hi! Let $S$ be a hyperbolic surface with metric $\rho$ and $N$ a hyperbolic $3$-dimensional manifold with bounded geometry. Let $g\colon (S,\rho)\to N$ be an incompressible pleated surface, that is ...
10
votes
3answers
619 views

Best known Margulis constants?

A Margulis number for a hyperbolic $n$-manifold $M=\mathbb{H}^n/\Gamma$ is a number $\epsilon>0$ such that for each $x\in\mathbb{H}^n$ the group generated by the elements in $\Gamma$ which move $x$ ...
4
votes
2answers
229 views

Pseudoanosov mapping torus and length of curves.

Let $M_{\phi}$ be a hyperbolic mapping torus coming from a pseudo-Anosov map $\phi$ in a surface $S$. Is there any way to estimate the length of the geodesic representing a given curve in the surface ...
4
votes
3answers
414 views

teichmuller geodesics and hyperbolic mapping torus

Given a pseudoanosov map $\phi$ of a surface $S$, there is a geodesic $\sigma$ in Teichmuller space (with the teichmuller metric) that is an axis for $\phi$, In other words, $\phi$ acts as a ...
10
votes
1answer
522 views

Measure on the Boundary of a Hyperbolic Group

Let $\Gamma$ be a non-elementary Gromov's $\delta$-hyperbolic group. Let $B(1,n)$ be the set of elements at distance at most $n$ from the identity and let $\partial B(1,n)$ be the elements at distance ...
4
votes
1answer
345 views

Characterization of the moduli space of the pair of pants in terms of the modules of the extremal ring domains

Hi, I was thinking about the following question ; I will appreciate it if somebody can give me a full or partial answer or can at least cite any reference(s)/ papers etc : By $ \bar{P} $ , we ...
4
votes
1answer
183 views

Rank of a group generated by side-pairing isometries of a polyhedron

Let $P$ be a compact convex polyhedron in $\mathbb{H}^3$. Let $G$ be a group generated by side-pairing isometries of $P$. Is there an algorithm to find the rank of $G$?
16
votes
2answers
714 views

Non-residually finite matrix groups

By Malcev's theorem, every finitely generated linear group is residually finite (RF). On the other hand, say, the group of rational numbers is linear, but is not residually finite. Thus, one has to ...
3
votes
1answer
349 views

Relationship between hyperbolicity in group theory and hyperbolicity in geometry

Could somebody teach me about the relationship, if any, between hyperbolicity in groups (in Gromov sense) and hyperbolicity in 3-dimensional orbifolds? To be more specific, let Q be a 3-dimensional ...
2
votes
1answer
499 views

The version of Montel's theorem used in the proof of Jenkins-Strebel differential

Hello, I am afraid that my main question might be a bit too elementary, but still I ask : In short, my question is "what is the version of Montel's theorem for a family of holomorphic maps from an ...
3
votes
0answers
545 views

Thales' Theorem for Hyperbolic Geometry [duplicate]

In Euclidean geometry Thales' Theorem says that if you view a diameter of a circle from any point on the perimeter it occupies exactly $90$ degrees in your field of view. More generally for any ...
24
votes
2answers
773 views

chromatic number of the hyperbolic plane

A notorious problem in combinatorics is the following: If we color $\mathbb{R}^2$ so that no pair of points at unit distance get the same color, what is the fewest number of colors required? This ...
10
votes
0answers
432 views

How does duality of symmetric spaces explain the hyperbolic cosine theorem?

There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $G/K$ is a symmetric space of noncompact type, $g=k+p$ the ...
5
votes
3answers
354 views

center of fundamental group of finite volume-hyperbolic orbifold

Is center of a fundamental group of finite volume-hyperbolic n-orbifold trivial? Is there a good reference that the proof is wriiten?
1
vote
0answers
162 views

Measuring the complexity of a knot by minimum number of simplices to tile the complement

This is essentially a duplicate of: Lower bound on number of tetrahedra needed to triangulate a knot complement Suppose a knot $K\subseteq\mathbb S^3$ is such that the complement $\mathbb ...
4
votes
1answer
197 views

Mapping torus relative to an infinite orbit can be hyperbolic with finite volume?

Consider a homeomorphism of the sphere with an infinite orbit converging forwards and backwards to the same point. Remove the orbit and the accumulation point and make the mapping torus. Can the ...
14
votes
1answer
598 views

Why is there a unique hyperbolic simplex of largest area?

Why is there a unqiue ideal $n$-simplex in $\mathbb H^n$ with largest volume for $n\geq 3$? For $n=3$, this is a standard calculation, and for larger dimensions is much harder (see Haagerup and ...
7
votes
1answer
483 views

Is there a simplicial volume definition of Chern Simons invariants?

Suppose we have some compact hyperbolic 3-manifold $M=\Gamma\backslash\mathbb H^3$. Now we know that the hyperbolic volume of $M$ can be defined as (a constant times) the simplicial volume of the ...
3
votes
2answers
371 views

volume of complex hyperbolic manifolds

I would like to know if there are in the literature explicit computations of the volume of complex hyperbolic manifolds. More precisely, let $\mathcal O$ be an imaginary quadratic number field, and ...
2
votes
1answer
171 views

Example of hyperbolic 3-fold with no embedded incompressible subsurfaces

Kahn-Markovic show that every hyperbolic 3-fold contains an immersed $\pi_1$ injective surface. Are there any known examples of hyperbolic 3-folds that do not contain a embedded $\pi_1$ injective ...
2
votes
2answers
312 views

Hurwitz's automorphisms theorem for infinite genus Riemann surfaces

Hurwitz's automorphisms theorem states that for a compact Riemann surface $X$ the cardinality of $Aut(X)$, the group of holomorphic automorphisms, is bounded above by $84(g(X)-1)$ and is therefore ...
9
votes
3answers
685 views

Primitive elements in a free group of rank three

It is well-known that the fundamental group of a twice-punctured torus is a free group of rank three. I see that there is no one-to-one correspondence between the homotopy classes of essential ...
3
votes
3answers
931 views

Poincare Metric on Hyperbolic Plane

as is well known, we can put a metric on the upper half plane $\mathbb{R}^+ \times \mathbb{R}$ by setting $$ d\left((x,t);(x',t')\right):=\log\left(\frac{1 + \delta}{1 - \delta}\right)^{1/2}, $$ ...
12
votes
2answers
657 views

How to see isometries of figure 8 knot complement

The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this ...
2
votes
1answer
866 views

How to rigorously prove that simple closed curves on a surface are primitive closed curves ?

Let me first state the definitions : A not-nullhomotopic closed curve / loop $c$ on an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ...
0
votes
1answer
510 views

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? [closed]

In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, i.e., $$ S_{\Delta}=\frac{1}{2}a.h, $$ where $a$ is the length of base and the $h$ is ...
4
votes
1answer
304 views

Arithmetic Fuchsian group

Dear all, I have the following questions: Are all Fuchsian groups of signature $(0;2,2,2,\infty)$ arithmetic? What is known about the trace fields of these groups? Best, K.
3
votes
2answers
248 views

In hyperbolic 3-orbifold with totally geodesic boundary case, is it true: rank(the fundamental group of boundary M)< or equal 2 rank(fundmental group of M)?

For a orientable three manifold M with totally geodesic boundary, this inequality is true. Because the rank of (fundemantal group of boundary M)=rank (homology group of boundary M ) then we use the ...
1
vote
1answer
103 views

Will the rank of fundemantal 3 manifold be decreased is I module the n(n>1) times of a element?

I am doing some rank control about the fundamental group of a 3 dim hyperbolic orbifold. After cuting out the regular neighborhood of all the singularity, I get a manifold with imcompressible ...
4
votes
1answer
438 views

The geometrical meaning of the common value in the law of sines in hyperbolic geometry

What is the geometrical meaning of the common value in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}$ in hyperbolic geometry? I know the meaning of this ...
9
votes
2answers
883 views

Questions on Thurston's earthquake flow

Here are some questions about the earthquake deformation of hyperbolic surface that I can't answer or find references. I briefly recall the settings. Let's fix a closed surface $S$ with genus $g\geq ...
6
votes
0answers
209 views

Does every hyperbolic 3 manifold with totally geodesic boundary has some finite covering space with more than one boundary component?

I am thinking about the question that: if we double a hyperbolic 3 manifold along its boundary, will the rank of fundemental group of the resulting closed manifold be strictly larger than before?\ The ...
6
votes
1answer
263 views

Can most 3 dimensional hyperbolic orbifolds with finite volume be covered by a hyperbolic manifold?

If G is a discrete cofinite volume subgroup of PSL(2,C),then G acts on H3, H3/G is a 3-dim hyperbolic orbifold N with finite volume, my question is : Is it right in most situations that we can find a ...
5
votes
2answers
294 views

Optimal pants decompositions of a hyperbolic surface

Let $S$ be a hyperbolic surface, which is not the punctured torus or $4$-holed sphere. I am interested in finding a ``geometrically optimal'' pants decomposition on $S$. Here is a candidate ...
2
votes
1answer
438 views

The smallest positive eigenvalue and the length of the shortest geodesic

I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two. Let $X$ ...
0
votes
1answer
211 views

Homotopy Groups of de Sitter and Anti-de Sitter?

Given $n$-dimensional de Sitter or Anti-de Sitter space, $dS_n$ or $AdS_n$, what are the homotopy groups $\pi_m(dS_n)$ and $\pi_m(AdS_n)$ and how does one calculate such things?
3
votes
1answer
465 views

Connection 1-forms of a Riemannian metric and the norm of the Hessian and ( seemingly ) two different definitions of Hessian and its norm

In the paper "On Quasiconformal Harmonic Maps " (link here) by L. F. Tam and T.Y.H. Wan, Pacific Journal of Mathematics, vol 182, no 2, 1998, in section 1, they define the Hessian of a function $f ...
1
vote
0answers
222 views

A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$

Hello, This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states : If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p ...
7
votes
3answers
984 views

Hyperbolicity on Riemann Surfaces

For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a ...
2
votes
1answer
641 views

A question on part of “An introduction to teichmuller spaces” by Imayoshi-Taniguchi

I was reading the part on Fenchel-Nielsen coordinates, the proof on page 64, I don't understand when they say: Since $\theta_j(t_1)=\theta_j(t_2)$, $j=1,\ldots,3g-3$, all $g_k$ can be glued ...
6
votes
1answer
271 views

What is the complex structure on the boundary torus of a hyperbolic knot complement?

Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary ...
8
votes
0answers
366 views

Is there a general dilogarithm formula for the Cheeger--Chern--Simons class?

I'm looking for a generalization of the calculation of the hyperbolic volume and Chern--Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm. ...
17
votes
3answers
796 views

Hyperbolic Coxeter polytopes and Del-Pezzo surfaces

Added. In the following link there is a proof of the observation made in this question: http://dl.dropbox.com/u/5546138/DelpezzoCoxeter.pdf I would like to find a reference for a beautiful ...
7
votes
1answer
458 views

Laminations as a limit of ideal triangulations

I am wondering about the following: Suppose that $S$ is a non-compact hyperbolic surface of finite area. Suppose that $\lambda \subset S$ is a non-trivial, geodesic, measured lamination. ...
1
vote
1answer
362 views

Two questions from Hubbard's Teichmuller theory book Vol I, P. 130 , Thm 4.4.1, ( QC maps )

I was studying Theorem 4.4.1 from John H. Hubbard's Teichmuller Theory, vol I, Theorem 4.4.1 ( P. 129 ) which states : Let $X,Y$ be two hyperbolic Riemann surfaces with hyperbolic metrics $d_X,d_Y$ ...
3
votes
2answers
395 views

a little question about Heegaard splitting

for any compact orientable hyperbolic 3-manifold with totally geodesic boundary, is there a strongly irreducible heegaard splitting?